Method and System for Arbitrary-Resolution Transforms of Frequency-Space and Inverse Frequency-Space Data

1. A method for data reconstruction or deconstruction, comprising: (a) acquiring, by inputting into a processing device or storing in a memory, a starting-space data set comprising a plurality of N-dimensional starting-space sample points selected by a user or a computer from an acquired image, audio, video, molecular analysis, chemical analysis, protein analysis, digital signal, digital data compression, or spectroscopy data superset, wherein each N-dimensional starting-space sample point comprises a set of associated sample-point component numbers, each set of sample-point component numbers notated herein as {k x , k y , k 3 , k 4 , . . . , k N ; a, b}, and wherein starting-space is either frequency-space or inverse frequency-space; (b) calculating, on a processor, for each of at least one user-selected or computer-selected x-coordinate p x in target-space, an associated horizontal key res1 comprising at least one horizontal key element, wherein target-space is frequency-space when starting-space is inverse frequency-space and is inverse frequency-space when starting-space is frequency-space, and wherein each horizontal key element is notated as res1 i and comprises at least one term of the sum: ∑ z = r 1 r 2 ⁢ ( ⅇ j2π ⁢ ⁢ ky i , z ⁢ E 1 ⁢ p x ⁢ E 2 * < a i , z , b i , z > ) wherein j 2 =−1, index z is a positive integer, and r 1 =1 and r 2 =L i , wherein L i is a number of k y values paired with at least one of the k x -notated values in the starting-space data set, and further wherein E 1 and E 2 are each a region scaling factor, ky i,z is one of the sample-point component numbers in the starting-space data set, <a i,z , b i,z > is one of a plurality of complex-number starting-space data values in the starting-space data set, and wherein * represents complex-number multiplication; (c) calculating, on the processor, for each of at least one user-selected or computer-selected target-space point (p x , p y ) comprising a current target-space x-coordinate p x and a current target-space y-coordinate p y , a color, wherein the color comprises at least one term of the sum: ∑ i = r 3 r 4 ⁢ ( ⅇ j2π ⁢ ⁢ p y ⁢ E 5 ⁢ kx i ⁢ E 6 * res ⁢ ⁢ 1 i ) wherein j 2 =−1, index i is a positive integer, r 3 =1 and r 4 =n, wherein n is a number of k x values in the starting-space data set, and further wherein E 5 and E 6 are each a region scaling factor, kx i is one of the sample-point component numbers in the starting-space data set, res1 i is one of the horizontal key elements associated with at least one of the horizontal keys, and wherein * represents complex-number multiplication; and (d) forming, by storing in the memory, outputting to a display device, storing on a storage device, or outputting to an output device, target-space data comprising at least one of the computed colors, thereby generating target-space data having a resolution corresponding to a number or density of the selected target-space points.

2. The method of claim 1 , further comprising applying at least one density-compensation function, by calculating on the processor or storing in the memory, at least one additional starting-space data value, wherein each additional starting-space data value comprises a complex number computed based on at least one sample-point component number in the starting-space data set.

3. The method of claim 2 , wherein one of the at least one density-compensation function is a ramp filter, Voronoi weighted filter, modified ramp filter, apodized ramp filter, Hanning/Hann filter, Butterworth filter, ramp-Hanning filter, ramp-Hamming filter, Parzen filter, Hamming filter, Shepp-Logan filter, Cosine filter, or square root of a ramp filter.

4. The method of claim 2 , wherein one of the at least one density-compensation function is defined as a function of the distance between at least two starting-space coordinates.

5. A system for reconstructing or deconstructing data, comprising: (a) a processing device configured to acquire, by way of an input interface, a starting-space data set comprising a plurality of N-dimensional starting-space sample points selected by a user or a computer from an acquired image, audio, video, molecular analysis, chemical analysis, protein analysis, digital signal, digital data compression, or spectroscopy data superset, wherein each N-dimensional starting-space sample point comprises a set of associated sample-point component numbers, each set of sample-point component numbers notated herein as {k x , k y , k 3 , k 4 , . . . , k N ; a, b}, and wherein starting-space is either frequency-space or inverse frequency-space; (b) a processor configured to calculate, for each of at least one user-selected or computer-selected x-coordinate p x in target-space, an associated horizontal key res1 comprising at least one horizontal key element, wherein target-space is frequency-space when starting-space is inverse frequency-space and is inverse frequency-space when starting-space is frequency-space, and wherein each horizontal key element is notated as res1 i and comprises at least one term of the sum: ∑ z = r 1 r 2 ⁢ ( ⅇ j2π ⁢ ⁢ ky i , z ⁢ E 1 ⁢ p x ⁢ E 2 * < a i , z , b i , z > ) wherein j 2 =−1, index z is a positive integer, and r 1 =1 and r 2 =L i , wherein L i is a number of k y values paired with at least one of the k x -notated values in the starting-space data set, and further wherein E 1 and E 2 are each a region scaling factor, ky i,z is one of the sample-point component numbers in the starting-space data set, <a i,z , b i,z > is one of a plurality of complex-number starting-space data values in the starting-space data set, and wherein * represents complex-number multiplication, and further configured to calculate, for each of at least one user-selected or computer-selected target-space point (p x , p y ) comprising a current target-space x-coordinate p x and a current target-space y-coordinate p y , a color, wherein the color comprises at least one term of the sum: ∑ i = r 3 r 4 ⁢ ( ⅇ j2π ⁢ ⁢ p y ⁢ E 5 ⁢ kx i ⁢ E 6 * res ⁢ ⁢ 1 i ) wherein j 2 =−1, index i is a positive integer, r 3 =1 and r 4 =n, wherein n is a number of k x values in the starting-space data set, and further wherein E 5 and E 6 are each a region scaling factor, kx i is one of the sample-point component numbers in the starting-space data set, res1 i is one of the horizontal key elements associated with at least one of the horizontal keys, and wherein * represents complex-number multiplication; and (c) a memory configured to form target-space data by storing at least one of the computed colors, thereby generating target-space data having a resolution corresponding to a number or density of the selected target-space points.

6. The system of claim 5 , wherein the memory further comprises being configured to apply at least one density-compensation function, by storing at least one additional starting-space data value, wherein each additional starting-space data value comprises a complex number computed based on at least one sample-point component number in the starting-space data set.

7. The system of claim 6 , wherein one of the at least one density-compensation function is a ramp filter, Voronoi weighted filter, modified ramp filter, apodized ramp filter, Hanning/Hann filter, Butterworth filter, ramp-Hanning filter, ramp-Hamming filter, Parzen filter, Hamming filter, Shepp-Logan filter, Cosine filter, or square root of a ramp filter.

8. The system of claim 6 , wherein one of the at least one density-compensation function is defined as a function of the distance between at least two starting-space coordinates.

9. A non-transitory computer-readable storage device storing instructions that when executed by a computer cause the computer to perform a method for using a computer system to reconstruct or deconstruct data, the method comprising: (a) acquiring, by inputting into a processing device or storing in a memory, a starting-space data set comprising a plurality of N-dimensional starting-space sample points selected by a user or a computer from an acquired image, audio, video, molecular analysis, chemical analysis, protein analysis, digital signal, digital data compression, or spectroscopy data superset, wherein each N-dimensional starting-space sample point comprises a set of associated sample-point component numbers, each set of sample-point component numbers notated herein as {k x , k y , k 3 , k 4 , . . . , k N ; a, b}, and wherein starting-space is either frequency-space or inverse frequency-space; (b) calculating, on a processor, for each of at least one user-selected or computer-selected x-coordinate p x in target-space, an associated horizontal key res1 comprising at least one horizontal key element, wherein target-space is frequency-space when starting-space is inverse frequency-space and is inverse frequency-space when starting-space is frequency-space, and wherein each horizontal key element is notated as res1 i and comprises at least one term of the sum: ∑ z = r 1 r 2 ⁢ ( ⅇ j2π ⁢ ⁢ ky i , z ⁢ E 1 ⁢ p x ⁢ E 2 * < a i , z , b i , z > ) wherein j 2 =−1, index z is a positive integer, and r 1 =1 and r 2 =L i , wherein L i is a number of k y values paired with at least one of the k x -notated values in the starting-space data set, and further wherein E 1 and E 2 are each a region scaling factor, ky i,z is one of the sample-point component numbers in the starting-space data set, <a i,z , b i,z > is one of a plurality of complex-number starting-space data values in the starting-space data set, and wherein * represents complex-number multiplication; (c) calculating, on the processor, for each of at least one user-selected or computer-selected target-space point (p x , p y ) comprising a current target-space x-coordinate p x and a current target-space y-coordinate p y , a color, wherein the color comprises at least one term of the sum: ∑ i = r 3 r 4 ⁢ ( ⅇ j2π ⁢ ⁢ p y ⁢ E 5 ⁢ kx i ⁢ E 6 * res ⁢ ⁢ 1 i ) wherein j 2 =−1, index i is a positive integer, r 3 =1 and r 4 =n, wherein n is a number of k x values in the starting-space data set, and further wherein E 5 and E 6 are each a region scaling factor, kx i is one of the sample-point component numbers in the starting-space data set, res1 i is one of the horizontal key elements associated with at least one of the horizontal keys, and wherein * represents complex-number multiplication; and (d) forming, by storing in the memory, outputting to a display device, storing on a storage device, or outputting to an output device, target-space data comprising at least one of the computed colors, thereby generating target-space data having a resolution corresponding to a number or density of the selected target-space points.

10. The computer-readable storage device of claim 9 , the method further comprising applying at least one density-compensation function, by calculating on the processor or storing in the memory, at least one additional starting-space data value, wherein each additional starting-space data value comprises a complex number computed based on at least one sample-point component number in the starting-space data set.

11. The computer-readable storage device of claim 10 , wherein one of the at least one density-compensation function is a ramp filter, Voronoi weighted filter, modified ramp filter, apodized ramp filter, Hanning/Hann filter, Butterworth filter, ramp-Hanning filter, ramp-Hamming filter, Parzen filter, Hamming filter, Shepp-Logan filter, Cosine filter, or square root of a ramp filter.

12. The computer-readable storage device of claim 10 , wherein one of the at least one density-compensation function is defined as a function of the distance between at least two starting-space coordinates.